Clifford parallel

The Fresnel–Arago laws are three laws which summarise some of the more important properties of interference between light of different states of polarization. Augustin-Jean Fresnel and François Arago, both discovered the laws, which bear their name.

The laws are as follows:^[1]

1. Two orthogonal, coherent linearly polarized waves cannot interfere.
2. Two parallel coherent linearly polarized waves will interfere in the same way as natural light.
3. The two constituent orthogonal linearly polarized states of natural light cannot interfere to form a readily observable interference pattern, even if rotated into alignment (because they are incoherent).

One may understand this more clearly when considering two waves, given by the form $\mathbf {E_{1}} (\mathbf {r} ,t)=\mathbf {E} _{01}\cos(\mathbf {k_{1}\cdot r} -\omega t+\epsilon _{1})$ and $\mathbf {E_{2}} (\mathbf {r} ,t)=\mathbf {E} _{02}\cos(\mathbf {k_{2}\cdot r} -\omega t+\epsilon _{2})$ , where the boldface indicates that the relevant quantity is a vector, interfering. We know that the intensity of light goes as the electric field squared (in fact, $I=\epsilon v<\mathbf {E} ^{2}>_{T})$ , where the angled brackets denote a time average), and so we just add the fields before squaring them. Extensive algebra ^[2] yields a result for the intensity of the resultant wave, namely that: $I_{12}=\mathbf {E_{01}\cdot E_{02}} \cos \delta$ where $\delta$ represents the interference term; the phase difference arising from a combined path length and initial phase-angle difference; $\delta =(\mathbf {k_{1}\cdot r-k_{2}\cdot r} +\epsilon _{1}-\epsilon _{2})$ .

At this point, substituting various correct values of $\delta$ yields laws 1 & 3 above.

With classical wave formulation for law 1 at absorption the polarisation axis are $Y=sin\alpha sin(\omega t)$ and $X=cos\alpha sin(\omega t+\phi )$ which combine to $I={\sqrt {Y^{2}+X^{2}}}$ . Here $\alpha$ is the angle between the incoming polarization and the (orthogonal) axis of the two polarisers and $\phi$ is the phase shift between both waves. According quantum mechanics the probability of absorption is $\int _{0}^{2\pi }\!I^{2}\,d\omega t=...=\pi$ , so independent of $\phi$ .